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Recent questions tagged vector-analysis
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1
TIFR ECE 2022 | Question: 5
Let $Q$ be a unit square in the plane with corners at $(0,0),(0,1),(1,0)$ and $(1,1)$. Let $B$ be a ball of radius $1$ in the plane centered at the origin $(0,0)$. Let $Q+B$ denote the set of all vectors in the plane of the form $v+w,$ where $v \in Q$ and $w \in B$. The area of $Q+B$ is: $5+\pi$ $4+\pi$ $3+\pi$ $2+\pi$ $1+\pi$
admin
asked
in
Vector Analysis
Nov 30, 2022
by
admin
43.6k
points
13
views
tifrece2022
vector-analysis
vector-in-planes
1
vote
0
answers
2
TIFR ECE 2022 | Question: 10
Find the vector which is closest (in Euclidean distance) to $\left(\begin{array}{lll}-1 & 1 & 1\end{array}\right)$ which can be written in the form \[a\left(\begin{array}{lll} 1 & 1 & 1 \end{array}\right)+b\left(\begin{array}{lll} 0 ... None of the above
admin
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in
Vector Analysis
Nov 30, 2022
by
admin
43.6k
points
12
views
tifrece2022
vector-analysis
vector-in-planes
1
vote
0
answers
3
TIFR ECE 2021 | Question: 10
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$. Let the real number $a_{1}^{*}$ be such that it solves the following optimization problem \[d_{1}=\min _{a_{1} \in \mathbb{R}}\left\|\vec{u}-a_{1} \vec{v}_{1}\right\|,\] where we denote the length ... $\left\|\vec{u}-\left(\vec{p}_{2}-\vec{p}_{1}\right)\right\|$ $0$
admin
asked
in
Calculus
Nov 30, 2022
by
admin
43.6k
points
12
views
tifrece2021
vector-analysis
vector-in-planes
1
vote
0
answers
4
TIFR ECE 2020 | Question: 15
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors such that $\vec{v}_{1}^{T} \vec{v}_{2}=0$. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such that they solve the following optimization problem \[ ... $\left\|\vec{v}_{*}\right\|^{2}-\|\vec{u}\|^{2}$ $0$ None of the above
admin
asked
in
Vector Analysis
Nov 30, 2022
by
admin
43.6k
points
7
views
tifrece2020
vector-analysis
vector-in-planes
1
vote
0
answers
5
TIFR ECE 2018 | Question: 5
Suppose $\vec{u}, \vec{v}_{1}, \vec{v}_{2} \in \mathbb{R}^{n}$ are linearly independent vectors. Let the pair of real numbers $\left(a_{1}^{*}, a_{2}^{*}\right)$ be such that they solve the following optimization problem \[d=\min _{a_{1}, a_{2} \in \mathbb{R}}\left\ ... $\left\|\vec{v}_{*}\right\|^{2}-\|\vec{u}\|^{2}$ None of the above
admin
asked
in
Vector Analysis
Nov 29, 2022
by
admin
43.6k
points
17
views
tifrece2018
vector-analysis
vector-in-planes
0
votes
1
answer
6
GATE ECE 2021 | Question: 1
The vector function $F\left ( r \right )=-x\hat{i}+y\hat{j}$ is defined over a circular arc $C$ shown in the figure. The line integral of $\int _{C} F\left ( r \right ).dr$ is $\frac{1}{2}$ $\frac{1}{4}$ $\frac{1}{6}$ $\frac{1}{3}$
Arjun
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in
Vector Analysis
Feb 20, 2021
by
Arjun
6.0k
points
446
views
gateec-2021
vector-analysis
vector-in-planes
0
votes
0
answers
7
GATE ECE 2021 | Question: 16
If the vectors $(1.0,\:-1.0,\:2.0)$, $(7.0,\:3.0,\:x)$ and $(2.0,\:3.0,\:1.0)$ in $\mathbb{R}^{3}$ are linearly dependent, the value of $x$ is __________
Arjun
asked
in
Vector Analysis
Feb 20, 2021
by
Arjun
6.0k
points
171
views
gateec-2021
numerical-answers
vector-analysis
vector-in-planes
0
votes
0
answers
8
GATE ECE 2021 | Question: 17
Consider the vector field $F\:=\:a_{x}\left ( 4y-c_{1}z \right )+a_y\left ( 4x + 2z\right )+a_{z}\left ( 2y +z\right )$ in a rectangular coordinate system $(x,y,z)$ with unit vectors $a_{x},\:a_{y}$ and $a_{z}$. If the field $F$ is irrotational (conservative), then the constant $c_{1}$ (in integer) is _________________
Arjun
asked
in
Vector Analysis
Feb 20, 2021
by
Arjun
6.0k
points
144
views
gateec-2021
numerical-answers
vector-analysis
vector-in-planes
0
votes
0
answers
9
GATE ECE 2021 | Question: 37
For a vector field $D=\rho\cos^{2}\:\varphi \:a_{\rho }+z^{2}\sin^{2}\:\varphi \:a_{\varphi }$ in a cylindrical coordinate system $\left ( \rho ,\varphi ,z \right )$ with unit vectors $a_{\rho },a_{\varphi }$ and $a_{z}$, the ... $\left ( \rho =3, 0\leq z\leq 2 \right )$ (rounded off to two decimal places) is ________________
Arjun
asked
in
Vector Analysis
Feb 20, 2021
by
Arjun
6.0k
points
72
views
gateec-2021
numerical-answers
vector-analysis
0
votes
0
answers
10
GATE ECE 2020 | Question: 1
If $v_{1},v_{2}, \dots ,v_{6}$ are six vectors in $\mathbb{R}^{4}$ , which one of the following statements is $\text{FALSE}$? It is not necessary that these vectors span $\mathbb{R}^{4}$. These vectors are not linearly independent. Any four of these vectors form a basis ... $\mathbb{R}^{4}$ , then it forms a basis for $\mathbb{R}^{4}$.
go_editor
asked
in
Vector Analysis
Feb 13, 2020
by
go_editor
1.9k
points
308
views
gate2020-ec
vector-analysis
1
vote
0
answers
11
GATE ECE 2020 | Question: 2
For a vector field $\overrightarrow{A}$, which one of the following is $\text{FALSE}$? $\overrightarrow{A}$ is solenoidal if $\triangledown \cdot \overrightarrow{A}=0.$ $\triangledown \times \overrightarrow{A}$ ...
go_editor
asked
in
Vector Analysis
Feb 13, 2020
by
go_editor
1.9k
points
300
views
gate2020-ec
vector-analysis
0
votes
0
answers
12
GATE ECE 2020 | Question: 24
The random variable $Y=\int_{-\infty }^{\infty }W\left ( t \right )\phi \left ( t \right )dt, \text{ where } \phi \left ( t \right )=\begin{cases} 1; & 5\leq t\leq 7 &\\ 0; & \text{otherwise} \end{cases}$ and $W(t)$ is ... noise process with two-sided power spectral density $S_{W}\left ( f \right )=3 W/Hz$, for all $f$. The variance of $Y$ is ________.
go_editor
asked
in
Vector Analysis
Feb 13, 2020
by
go_editor
1.9k
points
122
views
gate2020-ec
numerical-answers
vector-analysis
gausss-theorem
1
vote
1
answer
13
GATE ECE 2016 Set 3 | Question: 27
If the vectors $e_1=(1,0,2)$, $e_2=(0,1,0)$ and $e_3=(-2,0,1)$ form an orthogonal basis of the three-dimensional real space $\mathbb{R}^3$, then the vector $\textbf{u}=(4,3,-3)\in \mathbb{R}^3 $ can be expressed as $\textbf{u}=-$ ... \frac{2}{5}$e_1+3e_2+$\large\frac{11}{5}$e_3 \\$ $\textbf{u}=-$\large\frac{2}{5}$e_1+3e_2-$\large\frac{11}{5}$e_3$
Milicevic3306
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in
Vector Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
215
views
gate2016-ec-3
vector-analysis
0
votes
0
answers
14
GATE ECE 2016 Set 3 | Question: 28
A triangle in the $xy$-plane is bounded by the straight lines $2x=3y, \: y=0$ and $x=3$. The volume above the triangle and under the plane $x+y+z=6$ is _______
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in
Vector Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
91
views
gate2016-ec-3
numerical-answers
vector-analysis
vector-in-planes
0
votes
0
answers
15
GATE ECE 2016 Set 3 | Question: 50
A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of $4.0\:kHz$ and two-sided noise power spectral density ${\large\frac{\eta}{2}}=2.5\times10^{-5}Watt\:per\:Hz$. If information at the rate ... transmitted over this channel with arbitrarily small bit error rate, then the minimum bit-energy $E_b$ (in mJ/bit) necessary is _______
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Vector Analysis
Mar 28, 2018
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Milicevic3306
15.8k
points
71
views
gate2016-ec-3
numerical-answers
vector-analysis
gauss's-theorem
0
votes
0
answers
16
GATE ECE 2016 Set 2 | Question: 5
Consider the time-varying vector $\textbf{I}=\hat{x}15\cos(\omega t)+\hat{y}5\sin(\omega t)$ in Cartesian coordinates, where $\omega> 0$ is a constant. When the vector magnitude $\mid \textbf{I} \mid$ is at its minimum value, the angle $\theta$ that $\textbf{I}$ makes with the $x$ axis (in degrees, such that $ 0\leq \theta \leq 180)$ ________
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Vector Analysis
Mar 28, 2018
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Milicevic3306
15.8k
points
254
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gate2016-ec-2
numerical-answers
vector-analysis
0
votes
0
answers
17
GATE ECE 2016 Set 2 | Question: 55
A positive charge $q$ is placed at $x=0$ between two infinte metal plates placed at $x=-d$ and at $x=+d$ respectively. The metal plates lie in the $yz$ plane. The charge is at rest at $t=0$, when a voltage $+V$ is applied to the plate at $-d$ and ... that the charge $q$ takes to reach the right plate is proportional to $d/V$ $\sqrt{d}/V$ $d/\sqrt{V}$ $\sqrt{d/V}$
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Vector Analysis
Mar 28, 2018
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Milicevic3306
15.8k
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50
views
gate2016-ec-2
vector-analysis
0
votes
0
answers
18
GATE ECE 2016 Set 1 | Question: 29
The region specified by $\{ (\rho,\varphi,z):3 \leq\rho\leq 5,\frac{\pi}{8}\leq\varphi\leq\frac{\pi}{4}, \: 3\leq z\leq4.5\}$ in cylindrical coordinates has volume of _________
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Vector Analysis
Mar 28, 2018
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Milicevic3306
15.8k
points
80
views
gate2016-ec-1
numerical-answers
vector-analysis
0
votes
0
answers
19
GATE ECE 2016 Set 1 | Question: 50
An analog pulse $s(t)$ is transmitted over an additive white Gaussian noise (AWGN) channel. The received signal is $r(t) = s(t) + n(t)$, where $n(t)$ is additive white Gaussian noise with power spectral density $\frac{N_0}{2}$. The received signal is passed ... $E_s > E_h$ ; $SNR_{max}>\frac{2E_s}{N_0} \\ $ $E_s < E_h$ ; $SNR_{max}=\frac{2E_h}{N_0}$
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Vector Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
79
views
gate2016-ec-1
vector-analysis
gauss's-theorem
0
votes
0
answers
20
GATE ECE 2015 Set 3 | Question: 3
If $C$ is a circle of radius $r$ with centre $z_{0},$ in the complex $z$-plane and if $n$ is a non-zero integer, then $\oint _{C}\frac{dz}{(z-z_{0})^{n+1}}$ equals $2\pi n j$ $0$ $\frac{nj}{2\pi}$ $2\pi n$
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Complex Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
67
views
gate2015-ec-3
vector-analysis
0
votes
0
answers
21
GATE ECE 2015 Set 3 | Question: 4
Consider the function $g(t) = e^{-t}\sin(2\pi t)u(t)$ where $u(t)$ is the unit step function. The area under $g(t)$ is ______.
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Vector Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
141
views
gate2015-ec-3
numerical-answers
vector-analysis
0
votes
0
answers
22
GATE ECE 2015 Set 3 | Question: 29
A vector field $\textbf{D} = 2\rho^{2}\:\textbf{a}_{\rho} + z\: \textbf{a}_{z}$ exists inside a cylindrical region enclosed by the surfaces $\rho =1,z = 0$ and $z = 5.$ Let $S$ be the surface bounding this cylindrical region. The surface integral of this field on $S(∯_{S} \textbf{D.ds})$ is _______.
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Vector Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
points
85
views
gate2015-ec-3
numerical-answers
vector-analysis
0
votes
0
answers
23
GATE ECE 2015 Set 2 | Question: 49
A zero mean white Gaussian noise having power spectral density $\dfrac{N_{0}}{2}$ is passed through an LTI filter whose impulse response $h(t)$ is shown in the figure. The variance of the filtered noise at $t = 4$ is $\dfrac{3}{2}A^{2}N_{0} \\$ $\dfrac{3}{4}A^{2}N_{0} \\$ $A^{2}N_{0} \\$ $\dfrac{1}{2}A^{2}N_{0}$
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Vector Analysis
Mar 28, 2018
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Milicevic3306
15.8k
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70
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gate2015-ec-2
vector-analysis
gauss's-theorem
0
votes
0
answers
24
GATE ECE 2015 Set 1 | Question: 25
A vector $\overrightarrow{P}$ is given by $\overrightarrow{P} = x^3y\overrightarrow{a}_x - x^2y^2\overrightarrow{a}_y - x^2 yz \overrightarrow{a}_z$. Which one of the statements is TRUE? $\overrightarrow{P}$ is ... irrotational, but not solenoidal $\overrightarrow{P}$ is neither solenoidal, nor irrotational $\overrightarrow{P}$ is both solenoidal and irrotational
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Vector Analysis
Mar 28, 2018
by
Milicevic3306
15.8k
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66
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gate2015-ec-1
vector-analysis
0
votes
0
answers
25
GATE ECE 2015 Set 1 | Question: 54
The electric field intensity of a plane wave traveling in free space is given by the following expression $\textbf{E}(x,t)=\textbf{a}_y \: 24 \: \pi \: \: \cos(\omega t - k_0 x) \: \: (V/m)$ ... $x+y=1$. The total time-averaged power (in mW) passing through the square area is _____________.
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Vector Analysis
Mar 28, 2018
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Milicevic3306
15.8k
points
48
views
gate2015-ec-1
numerical-answers
vector-analysis
0
votes
0
answers
26
GATE ECE 2015 Set 1 | Question: 55
Consider a uniform plane wave with amplitude $(E_0)$ of $10 \: V/m$ and $1.1$ GHz frequency travelling in air, and incident normally on a dielectric medium with complex relative permittivity $(\varepsilon _r)$ ... electric field component (in V/m) after it has travelled a distance of $10$ cm inside the dielectric region is ____________.
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Vector Analysis
Mar 28, 2018
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Milicevic3306
15.8k
points
60
views
gate2015-ec-1
numerical-answers
vector-analysis
0
votes
0
answers
27
GATE ECE 2014 Set 4 | Question: 2
The magnitude of the gradient for the function $f(x,y,z) = x^2 +3y^2 +z^3$ at the point $(1,1,1)$ is ___________.
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Vector Analysis
Mar 26, 2018
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Milicevic3306
15.8k
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47
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gate2014-ec-4
numerical-answers
vector-analysis
gradient
0
votes
0
answers
28
GATE ECE 2014 Set 4 | Question: 3
Let $X$ be a zero mean unit variance Gaussian random variable. $E[ \mid X \mid ]$ is equal to __________
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Vector Analysis
Mar 26, 2018
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Milicevic3306
15.8k
points
60
views
gate2014-ec-4
numerical-answers
vector-analysis
gausss-theorem
random-variable
0
votes
0
answers
29
GATE ECE 2014 Set 4 | Question: 5
The directional derivative of $f(x,y)= \frac{xy}{\sqrt{2}} (x+y)$ at $(1,1)$ in the direction of the unit vector at an angle of $\frac{\pi}{4}$ with $y$-axis, is given by _________.
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Vector Analysis
Mar 26, 2018
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Milicevic3306
15.8k
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49
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gate2014-ec-4
vector-analysis
numerical-answers
0
votes
0
answers
30
GATE ECE 2014 Set 4 | Question: 22
If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a fixed time interval will be Poisson Gaussian Exponential Gamma
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in
Vector Analysis
Mar 26, 2018
by
Milicevic3306
15.8k
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101
views
gate2014-ec-4
vector-analysis
gauss's-theorem
0
votes
0
answers
31
GATE ECE 2014 Set 4 | Question: 49
Consider a communication scheme where the binary valued signal $X$ satisfies $P\{X=+1\}=0.75$ and $P\{X=-1 \}=0.25$. The received signal $Y=X+Z$, where $Z$ is a Gaussian random variable with zero mean and variance ... $\sigma ^2$
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Vector Analysis
Mar 26, 2018
by
Milicevic3306
15.8k
points
49
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gate2014-ec-4
vector-analysis
gauss's-theorem
0
votes
0
answers
32
GATE ECE 2014 Set 4 | Question: 52
Consider a discrete-time channel $Y=X +Z$, where the additive noise $Z$ is signal-dependent. In particular, given the transmitted symbol $ X \in \{-a , +a\}$ at any instant, the noise sample $Z$ is chosen independently from a Gaussian distribution with mean $\beta X$ and unit ... $\beta = -0.3$, the BER is closest to $10^{-7}$ $10^{-6}$ $10^{-4}$ $10^{-2}$
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Vector Analysis
Mar 26, 2018
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Milicevic3306
15.8k
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48
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gate2014-ec-4
vector-analysis
gausss-theorem
0
votes
0
answers
33
GATE ECE 2014 Set 4 | Question: 54
Gven $\overrightarrow{F} = z \hat{a}_x + x \hat{a}_y + y \hat{a}_z$. If $S$ represents the portion of the sphere $x^2 +y^2+z^2=1$ for $z \geq 0$, then $\int _s \nabla \times \overrightarrow{F} \cdot \overrightarrow{ds}$ is __________.
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Vector Analysis
Mar 26, 2018
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Milicevic3306
15.8k
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132
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gate2014-ec-4
numerical-answers
vector-analysis
0
votes
0
answers
34
GATE ECE 2014 Set 3 | Question: 53
Given the vector $\textbf{A}= ( \cos x ) ( \sin y )\hat{a_{x}}+( \sin x )( \cos y )\hat{a_{y}},$ where $\hat{a_{x}},$ $\hat{a_{y}}$ denote unit vectors along $x$, $y$ directions, respectively. The magnitude of curl of $\textbf{A}$ is __________
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Vector Analysis
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Milicevic3306
15.8k
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73
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gate2014-ec-3
numerical-answers
vector-analysis
0
votes
0
answers
35
GATE ECE 2014 Set 2 | Question: 22
The capacity of a band-limited additive white Gaussian noise (AWGN) channel is given by $C = W \log_{2}\left ( 1+\frac{p} {\sigma ^{2}w} \right )$ bits per second (bps), where $W$ is the channel bandwidth, $P$ is the average power received ... channel capacity (in kbps) with infinite bandwidth $(W\rightarrow \infty )$ is approximately $1.44$ $1.08$ $0.72$ $0.36$
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Vector Analysis
Mar 26, 2018
by
Milicevic3306
15.8k
points
95
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gate2014-ec-2
vector-analysis
gauss's-theorem
0
votes
0
answers
36
GATE ECE 2014 Set 2 | Question: 29
If $\overrightarrow {r}= x\hat{a_{x}}+y\hat{a_{y}}+z\hat{a_{z}}$ and $\mid \overrightarrow{r} \mid= r$ , then $\text{div} ( r^{2} \nabla ( \text{ln }r ) )$ = _______ .
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Vector Analysis
Mar 26, 2018
by
Milicevic3306
15.8k
points
52
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gate2014-ec-2
vector-analysis
numerical-answers
0
votes
0
answers
37
GATE ECE 2014 Set 1 | Question: 28
The volume under the surface $z(x,y) = x + y$ and above the triangle in the $x – y$ plane defined by $\{0 \leq y \leq x \: \text{and} \: 0 \leq x \leq 12\}$ is _______.
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Vector Analysis
Mar 26, 2018
by
Milicevic3306
15.8k
points
71
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gate2014-ec-1
numerical-answers
vector-analysis
0
votes
0
answers
38
GATE ECE 2014 Set 1 | Question: 46
Consider the state space model of a system, as given below ... The system is controllable and observable uncontrollable and observable uncontrollable and unobservable controllable and unobservable
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in
Vector Analysis
Mar 26, 2018
by
Milicevic3306
15.8k
points
59
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gate2014-ec-1
vector-analysis
0
votes
0
answers
39
GATE ECE 2014 Set 1 | Question: 53
In spherical coordinates, let $\hat{a_{\theta}},\hat{a_{\phi}}$ denote unit vectors along the $\theta,\phi$ directions. $\textbf{E} = \dfrac{100}{r}\sin\theta \cos (\omega t - \beta r)\hat{a_{\theta}}\: V/m$ ... free space. The average power $(W)$ crossing the hemispherical shell located at $r = 1\:km,0\leq \theta \leq \pi/2$ is ______.
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Vector Analysis
Mar 26, 2018
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Milicevic3306
15.8k
points
84
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gate2014-ec-1
numerical-answers
vector-analysis
0
votes
0
answers
40
GATE ECE 2013 | Question: 52
A monochromatic plane wave of wavelength $\lambda = 600 \mu m$ is propagating in the direction as shown in the figure below. $\vec{E_{i}},\vec{E_{r}},$ and $\vec{E_{t}}$ ...
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Vector Analysis
Mar 26, 2018
by
Milicevic3306
15.8k
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81
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gate2013-ec
vector-analysis
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